Article

Ideal convergent subseries in Banach spaces

Published in: Quaestiones Mathematicae
Volume 42 , issue 6 , 2019 , pages: 765–779

DOI: 10.2989/16073606.2018.1497724

Author(s): Marek Balcerzak Institute of Mathematics, Poland , Michał Popławski Institute of Mathematics, Poland , Artur Wachowicz Institute of Mathematics, Poland

Keywords: 40A05 , 46B25 , 54E52 , 28A05 , 40A05 , 46B25 , 54E52 , 28A05

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Abstract

Assume that is an ideal on ℕ, and ∑_n x_n is a divergent series in a Banach space X. We study the Baire category, and the measure of the set A() := {t ∈ {0, 1}^ℕ: ∑_n t(n)x_n is -convergent}. In the category case, we assume that has the Baire property and ∑_n x_n is not unconditionally convergent, and we deduce that A() is meager. We also study the smallness of A() in the measure case when the Haar probability measure λ on {0, 1}^ℕ is considered. If is analytic or coanalytic, and ∑_n x_n is -divergent, then λ(A()) = 0 which extends the theorem of Dindoš, Šalát and Toma. Generalizing one of their examples, we show that, for every ideal on ℕ, with the property of long intervals, there is a divergent series of reals such that λ(A(Fin)) = 0 and λ(A()) = 1.

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